Objective

Complex dynamics studies the evolution of a complex manifold under the action of a holomorphic map. In this proposal we study the dynamical systems generated by transcendental (either entire or meromorphic) maps acting on the complex plane. By using a wide range of classic and new techniques, we investigates epecially the combinatorics of these maps: that is to say, we build relations between the dynamics of the transcendental map on some specific subset of the complex plane and the dynamics of the shift map on the space of infinite sequences over the integers. Combinatorics in this setting is a powerful tool to understand the dynamics of transcendental maps and to understand the structure of specific families of transcendental maps. The study of combinatorics for transcendental maps is also likely to offer new insights in the combinatorics for rational maps and possibly in other areas of complex dynamical systems, like the systems generated by the iteration of holomorphic maps on manifolds with more than one complex dimension.

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Minimising the complexity of complex dynamical systems

While the general concepts of static and dynamic are relatively easy to understand, when it comes to complex dynamical systems and high-level mathematics things become – well – complex. A new theorem enables mathematicians to focus on less information to extract most of the behaviour of dynamical systems.

Formally, a dynamical system is one whose state evolves with time over a ‘state space’. The state space is also called a topological manifold, the multi-dimensional equivalent of a curved surface. For example, a circle is a 1D manifold (a line) embedded in two dimensions, where each arc of the circle locally represents a line.
While a dynamical system is defined by a state space of integers or real numbers, a complex dynamical system has a corresponding complex manifold. With the support of the Marie Skłodowska-Curie programme, CoTraDy set out to study dynamical systems generated by 1D transcendental maps (non-polynomial such as exponential or trigonometric) acting on the complex plane.

A mathematical journey in time – and state space

According to project fellow Anna Miriam Benini, now of the University of Parma, and project coordinator Nùria Fagella of the University of Barcelona: “We were planning to investigate the consequences that a given combinatorics has for the dynamical behaviour of a given map.” Combinatorics is a sort of code from which one can theoretically determine all the dynamical features of a given map. Combinatorics enables mathematicians to group maps into classes in which all members have similar dynamics. Then, conversely, they can gain information about the dynamics of a map by knowing to which class of maps it belongs and its combinatorics. However, recovering the dynamics from the combinatorics is not always possible.

Finding stability in chaos

Benini continues: “What we managed to do was to link the combinatorics to the behaviour of a special set of points (called singular values) which are responsible for most of the dynamics of the map itself. As a result, we were able to obtain information about the equilibrium states of the dynamical system (the periodic points) and their relation to these points.” In other words, the team further simplified the ‘code’ or, rather, its interpretation by being able to extract the bulk of information about the dynamical behaviour from a subset of points without having to ‘connect all the dots’.

A dynamic duo of female mathematicians powers innovation

The work of Benini and Fagella provides important insight into combinatorics related to equilibrium states where the dynamical system returns over time. It could shed light beyond as well. Benini explains: “CoTraDy’s outcomes could open the door to better understanding of wandering domains, among the least understood phenomena in transcendental dynamics. These large sets of points move all together but never come back to themselves.” In addition, Benini is currently collaborating with others on the application of CoTraDy’s techniques to 2D transcendental dynamics, a broad field in which little is known.
Fagella concludes: “Our proof extends previous results and also gives a direct approach to understanding the relation between combinatorics, singular values, and equilibrium points.” Complex dynamical systems in any dimension are often complexifications of real ones, sometimes motivated by models of real-world dynamical systems like population change or the stock market. CoTraDy’s outcomes could help us look at these real models from a complex point of view to explain phenomena that are not understandable otherwise.

The topic of this project is in Dynamical System, a wide and variegated area of mathematics concerned with the study of the evolution of systems of any kind, from biological models to abstract mathematical systems.

The main purpose of this project was to study a specific type of abstract system given by the iteration of some functions called entire transcendental maps, and to deepen our understanding of such systems by investigating specific subsystems (the 'curves escaping points', or 'rays') which can be correlated with other simple and well understood examples. This type of approach takes the name of 'combinatorial study'.

This is a project in pure mathematics, so its direct impact for society in terms of research is long-term and difficult to predict. Other works concerning the dynamics of entire functions have proven useful in improving Newton's Method, a widely used algorithm to find roots of polynomials which has applications in several areas of science. As outreach activity we have planned and executed several conferences about fractals for high school students, which have been very successful. We believe that presenting female researchers in science to this particular type of public is important to promote female role models in science to teenagers, and encourages female students to pursue careers in STEM.

The scientific objectives were to investigate the patterns arising from the aforementioned subsystems (the curves escaping points, or rays) and their relation to periodic points, that is, equilibrium states of the systems.

For entire transcendental maps, periodic rays are special curves in the plane consisting of points whose orbits converge to infinity, and are one of the main objects of investigation of this project. Singular values are special values near which the function is not locally invertible, and periodic points are points that are invariant under some iterate of the function. With N. Fagella we proved that under general hypothesis periodic rays divide the plane into regions which, in a precise sense, encode the orbits of singular values, and that the latter are forced to interact with the nonrepelling periodic points present in such regions. This led to a new proof of the Fatou-Shishikura inequality, which in addition gives results for functions with infinitely many singular values. We have also been able to show that periodic point which are not the landing point of a periodic ray must interact in a precise way with singular orbits. With L. Rempe-Gillen we have been able to prove that, if the orbits of singular values are bounded, then all repelling periodic points are landing points of rays. With H. Peters and JE Fornaess we studied the entropy of entire transcendental functions, and with both of them and L. Arosio we extendend some results to a special class of transcendental automorphisms of C^2.With N. Fagella, Gwyneth Stallard, Phil Rippon and Vasso Evdoridou we produced a rather complete classification of bounded simply connected wandering domains as well as a series of original examples illustrating the classification.We submitted 4 preprints and published 2 additional papers. The results were presented at several international conferences and dynamical systems seminars.

We studied several new aspects of the dynamics of transcendental functions, especially concerning the relation between the set of escaping points and the set of periodic points and of singular values. We introduced two main new techniques in the field of transcendental dynamics: the concept of fundamental tails and a new way to obtain information about the dynamics of a transcendental function via the landing structure of its periodic rays. This technique has already found application in two preprints [EFJS], [PRS]. The other type of techniques which have been developed exploit results from [BF15]. This project extended unexpectedly also to dynamics in several complex variables, leading to the completion of a work on transcendental Henon maps of C^2 [ABFP] and a new work in progress which deals more specifically with escaping points of transcendental Henon maps.At the networking level, this project contributed to strengthen the collaboration between the research group from the Universitat de Barcelona and the research group of the Open University (UK), increasing international collaboration. It also contributed to strengthening the interaction between the field of dynamics in one complex variable and the field of dynamics in several complex variable, an interaction which we believe should be much stronger than it currently is.